How many such $5\text{s}$ are there in the following number sequence, each of which is immediately preceded by $3$ or $4$ but not immediately followed by $8$ or $9$?
$$\text{3 5 9 5 4 5 5 3 5 8 4 5 6 7 3 5 7 5 5 4 5 2 3 5 1 0}$$

We need to find patterns of $35$ or $45$ where the character following $5$ is NOT $8$ or $9$. Let us evaluate each $5$ in the sequence step by step:
1. $3,5,9⟹$ Preceded by $3$, but followed by $9$ (Invalid).
2. $9,5,4⟹$ Not preceded by $3$ or $4$ (Invalid).
3. $4,5,5⟹$ Preceded by $4$, followed by $5$ (Valid - Count = 1).
4. $5,5,3⟹$ Not preceded by $3$ or $4$ (Invalid).
5. $3,5,8⟹$ Preceded by $3$, but followed by $8$ (Invalid).
6. $4,5,6⟹$ Preceded by $4$, followed by $6$ (Valid - Count = 2).
7. $3,5,7⟹$ Preceded by $3$, followed by $7$ (Valid - Count = 3).
8. $7,5,5⟹$ Not preceded by $3$ or $4$ (Invalid).
9. $5,5,4⟹$ Not preceded by $3$ or $4$ (Invalid).
10. $4,5,2⟹$ Preceded by $4$, followed by $2$ (Valid - Count = 4).
11. $3,5,1⟹$ Preceded by $3$, followed by $1$ (Valid - Count = 5).

There are $5$ such valid $5\text{s}$.